The difference between DISTINCT and GROUP BY in Mysql is that they usually generate the same query plan, so performance should be the same across both query constructs.
GROUP BY should be used to apply aggregate operators to each group. If all you need is to remove duplicates then use DISTINCT. If you are using sub-queries execution plan for that query varies so in that case you need to check the execution plan before making decision of which is faster.
Example of DISTINCT:
SELECT DISTINCT Employee, Dept FROM Employees
Example of GROUP BY:
Dept FROM Employees GROUP BY Employee,
Example of GROUP BY with aggregate function:
Dept, COUNT(*) EmployeeCount FROM Employees GROUP BY Employee,